Radiative corrections in the Zeeman effect of 23 P states of helium

In order to resolve the long-standing discrepancy betw theory and experiment for the Zeeman coupling factor gL8 of the atomic 2PJ states of helium@1#, high-precision calculations, including relativistic corrections, have been done cently by several authors @2,3#. Their results fordgL5gL8 2gL with gL512me /mN are 10.6310 , 8.838310, and 10.719310 in Refs. @1#, @2#, and @3# respectively, which should be compared with the best experimental re @4# dgL54.9(2.9)310 . Anthony and Sebastian @2#, extending the work of othe authors@5,6#, also included radiative corrections, and fou by means of an accurate calculation that these correct provide a contribution of 1.79 310 to the orbitalgL8 factor, this contribution being too small to resolve the discrepan Such corrections come from the term proportional to eBW 0 3rW•AW vac, which is part of e AW 2 with AW 5(1/2)BW 03rW 1AW vac, whereBW 0 is the external magnetic field and AW vac is the potential vector of the vacuum field. The purpose of this paper is to show that there are o radiative corrections, not considered in previous wo coming from terms containingep W •AW 5(1/2)eB W 03rW•pW 1ep W •AW vac, in third order of perturbation, but of the sam order as those mentioned above @2#. It is also shown that the radiative corrections coming from ( 2e/m)SW •BW vac are zero or negligible.


I. INTRODUCTION
In order to resolve the long-standing discrepancy between theory and experiment for the Zeeman coupling factor g L Ј of the atomic 2 3 P J states of helium ͓1͔, high-precision calculations, including relativistic corrections, have been done recently by several authors ͓2,3͔.Their results for ␦g L ϭg L Ј Ϫg L with g L ϭ1Ϫm e /m N are 10.6ϫ10Ϫ6 , 8.838ϫ10 Ϫ6 , and 10.719ϫ10 Ϫ6 in Refs.͓1͔, ͓2͔, and ͓3͔ respectively, which should be compared with the best experimental result ͓4͔ ␦g L ϭ4.9(2.9)ϫ10Ϫ6 .
Anthony and Sebastian ͓2͔, extending the work of other authors ͓5,6͔, also included radiative corrections, and found by means of an accurate calculation that these corrections provide a contribution of 1.79ϫ10 Ϫ7 to the orbital g L Ј factor, this contribution being too small to resolve the discrepancy.Such corrections come from the term proportional to e 2 B ជ 0 ϫr ជ •A ជ vac , which is part of e 2 A ជ 2 with A ជ ϭ(1/2)B ជ 0 ϫr ជ ϩA ជ vac , where B ជ 0 is the external magnetic field and A ជ vac is the potential vector of the vacuum field.
The purpose of this paper is to show that there are other radiative corrections, not considered in previous work, coming from terms containing ep ជ •A ជ ϭ(1/2)eB ជ 0 ϫr ជ •p ជ ϩep ជ •A ជ vac , in third order of perturbation, but of the same order as those mentioned above ͓2͔.It is also shown that the radiative corrections coming from (Ϫe/m)S ជ •B ជ vac are zero or negligible.

II. THEORY
We consider the Zeeman effect of the helium atom adopting the LS coupling scheme.The Zeeman Hamiltonian is given by where eϭϪ͉e͉, B ជ 0 is the external magnetic field, and the values for the orbital and spin gyromagnetic factors are as-sumed, with small error for the present calculations, to be 1 and 2, respectively.We consider the total Hamiltonian where H 0 and H rad are the Hamiltonians of the free atom and free radiation, respectively.The third term is the Zeeman Hamiltonian ͑1͒, and H A and H B account for the radiative corrections originating from the interaction of the two electrons of the atom with the vacuum field, where the scalar products are in a six-dimensional space taking into account the two electrons denoted by the indexes 1, and 2, that is, where B ជ vac (r ជ i ) is the vacuum magnetic field and A ជ vac (r ជ i ) is the potential vector of the vacuum field, expressed as a plane-wave expansion.
We use a perturbative treatment in which HЈϵH Z ϩH A ϩH B is considered a perturbation to H 0 ϩH rad , and we calculate the third order-energy correction.If the unperturbed state is the atomic state with energy E 0 and radiation state of zero photons, ͉0͘, the energy correction is *Electronic address: igonzalo@eucmax.sim.ucm.es where ͉n͘ and ͉nЈ͘ denote any atomic eigenstate of H 0 , J 2 , L 2 , S 2 , and J z ; ͉k ជ , ⑀ ជ ͘ and ͉k ជ Ј,⑀ ជ Ј͘ denote single photon states of wave vectors k ជ and k ជ Ј and polarizations ⑀ ជ and ⑀ ជ Ј with frequencies ϭck and ЈϭckЈ respectively.Several contributions appear in Eq. ͑6͒ of the form

͑7͒
where each H i , H j , and H l may be H Z , H A , and H B ͑note that the sum labeled k ជ ,⑀ ជ , may eventually contain the vacuum state͒.
Retaining in Eq. ͑6͒ only the terms proportional to e 3 and linear in B 0 , we obtain  (3) are then obtained from , where ͉ B ͉ϵ͉e͉ប/(2m).The states ͉͘ here considered are the following atomic states of helium: where v denotes the electronic configuration.The calculations of the different terms of ͑8͒ are grouped according to their similarity in different sections.

III. CORRECTION FROM THE TERM E AZA
The first term of Eq. ͑8͒ is given by

A. Low frequencies
Let us first consider the region of frequencies where the dipole approximation holds (e ik ជ •r ជ j Ӎ1); that is, р c ϵ␣mc 2 /ប.To simplify the calculation we decompose the integral as follows.The function of in the first term of Eq. ͑12͒ can be written and similarly, in the second term of Eq. ͑12͒, ͑14͒ Expression ͑12͒ can then be written as where

͑16͒
͑the summation over nЈ has been performed͒, and . In order that L z ϩ2S z ϭJ z ϩS z acts on the state , we use the commutator ͪ.

͑18͒
In this expression, it is easy to see that there is no contribution from J z since is an eigenstate of J z , so the terms with J z are equal and cancel.Analogously there is no contribution from S z if is an eigenstate of S z , which occurs for the state ͉ a ͘ϵ͉ 3 P 2 ,M J ϭ2͘ given by Eq. ͑9͒.
For the state ͉ b ͘ϵ͉ 3 P 1 ,M J ϭ1͘, given by Eq. ͑10͒, which is not an eigenstate of S z , there is also no contribution from S z , as can be shown inserting where we must note that S z connects only states with the same electronic configuration and the same quantum number M J .In our case, with M J ϭ1, JЈ can be JЈϭ1 and 2. For JЈϭ1, the inserted state is the same as b and the first term in Eq. ͑18͒ cancels the second one.For JЈϭ2, the product of the first two matrix elements in Eq. ͑19͒, summed over polarizations and integrated over angles, is where the Wigner-Eckart theorem has been applied using p ជ •p ជ ϭ p 0 p 0 ϩ p Ϫ1 p Ϫ1 ϩ p 1 p 1 and the orthogonality of the Clebsch-Gordan coefficients.Hence there is also no contribution from S z for ͉ 3 P 1 ,M J ϭ1͘ in Eq. ͑18͒, which becomes •B ជ 0 /B 0 , and rear- ranging the vector product, we obtain where we recall that p x ϭp 1x ϩp 2x , p y ϭp 1y ϩp 2y and and ͉n͘ represent antisymmetrized functions.Due to the indistinguishability of the electrons, ͉͗p 1x ͉n͘ϭ͉͗p 2x ͉n͘, and there are then four equal terms, which allows us to write Eq. ͑21͒ as

͑22͒
This expression is similar to that calculated by Anthony and Sebastian ͓2͔ for the radiative correction in the second order of perturbation.The integral in frequencies is , then the electron is in the continuum and, by conservation of momentum n Ӎបk 2 /(2m) Ӷ so that ӷ c , contrary to our assumption͔.Equation ͑22͒then becomes

͑23͒
where the sum over n has been made in the first term and its complex conjugate, which now cancel between them since ͗p 1x p 1y ͘ ϭ͗p 1y p 1x ͘ , leading to

͑24͒
which is independent of the cutoff frequency and equal to half of the quantity calculated by Anthony and Sebastian ͓2͔.
Therefore, the corresponding contribution to g L (3) , choosing The mentioned authors multiply by 3 4 the value they found to correct the self-energy contributions, obtaining the value ( 3 4 )2.39ϫ10Ϫ7 ϭ1.79ϫ10 Ϫ7 mentioned in Sec.I.

Concerning the term E AZA II
given by Eq. ͑17͒, it can be seen that the integral in frequencies is convergent, which suggests small contribution from high frequencies.We can also suppose here that n Ӷ c .The integral in frequencies in the first term of Eq. ͑17͒ is then It must be noted that ͉n͘ and ͉nЈ͘, connected by L z ϩ2S z ϭJ z ϩS z in the first term of Eq. ͑17͒, must have the same configuration, their maximum energetic difference being only due to the spin-orbit interaction, which is small.Hence, if we consider that ⑀ϵ( n / n Ј Ϫ1) is much smaller than unity, we can write ln( n / n Ј )ϵln(1ϩ⑀)Ӎ⑀ϩ⑀ 2 /2ϩ in Eq. ͑17͒ can be easily calculated, and it is close to 1 ͑assuming that n Ӷ c ͒.These results permit us to sum over n and nЈ in Eq. ͑17͒, which becomes where we have taken into account that the commutator ͓J z ϩS z ,p ជ •⑀ ជ ͔ϭiប( p ជ ϫ⑀ ជ ) z and that the sum over polarizations and the angular integration of the expression ( p ជ Since is an eigenstate of J z , it is obvious that the contribution from J z in Eq. ͑27͒ cancels.It can be seen that the contribution from S z in the same expression is zero.This is obvious for the state ͉ a ͘ϵ͉ 3 P 2 ,M J ϭ2͘, which is an eigenstate of S z .For the state ͉ b ͘ϵ͉ 3 P 1 ,M J ϭ1͘, given by Eq. ͑10͒, it can be seen in Eq. ͑27͒ that, once the polarizations are summed and the angular integration performed, taking into account that ͗p 2 ͘ M L ϭ0 ϭ͗p 2 ͘ M L ϭ1 , the expression cancels.Therefore the part of order ⑀ of E AZA II is zero. If we consider the second term of the expansion ln(1ϩ⑀) 2 ), the corresponding contribution to E AZA I is negligible since the maximum value of ( n Ϫ n Ј ) corresponds to spin-orbit interaction.We then have ⑀ϵ( n Ϫ n Ј )/ n Ј Ϸ␣ 2 ͑as is known from the fine structure theory͒ and ⑀ 2 Ϸ␣ 4 , which would lead to a correction term much lower than E AZA I , and is therefore negligible.Terms of order ⑀ 3 and higher will be even smaller.

B. High frequencies
Let us now analyze the contribution from high frequencies ͑above c ) to the term E AZA without using the dipole approximation.We write Eq. ͑13͒ as and, similarly, Eq. ͑14͒ as We have now that уmc 2 ␣/ប and then n , n Ј ϳmc 2 ␣ 2 /បӶ.͑For very high frequencies ϳmc 2 /ប, this is not true, but in this regime the electron becomes relativistic and could not be treated with our methods.However, such frequencies are effectively cut off due to the rapid convergence of the integral in frequencies which will be consid-ered͒.It is then easy to see that the terms in parentheses in Eqs.͑28͒ and ͑29͒ will give a contribution of order ␣ with respect to the previous one.We retain only the first term, 1/, which is the only one that may make a non-negligible contribution for our purpose.In this case, the sum over n and nЈ in Eq. ͑12͒ can be performed, and we obtain

͑30͒
In this expression, the terms coming from a process where the photon is emitted and absorbed by the same electron ͑i.e., jϭt͒ make a quadratically divergent contribution, which means that the nonrelativistic approximation fails.Actually these terms, in which only one electron is involved, contribute to the anomalous magnetic moment of the electron, and we think that they should not be considered in our calculation.Taking only the terms where different electrons contribute, the integral in is convergent.Rearranging the terms remembering that L z ϭl 1z ϩl 2z and S z ϭs 1z ϩs 2z , for the operator in the first matrix element of Eq. ͑30͒ we obtain In the above expression, only the even part in k ជ ͓proportional to cos(k ជ •r ជ 12 ) with r ជ 12 ϵr ជ 1 Ϫr ជ 2 ] contributes to the angular . Ex- pression ͑30͒ can then be written as which is zero when is an eigenstate of L z ϩ2S z ϭJ z ϩS z , which occurs for the state ͉ a ͘ϵ͉ 3 P 2 ,M J ϭ2͘.For the state ͉ b ͘ϵ͉ 3 P 1 ,M J ϭ1͘ given by Eq. ͑10͒, an eigenstate of J z but not of S z , the contribution-to g S (3) -is also zero due to the rotational invariance of N and its independence on the spin, which leads to ͗N͘ M L ϭ1 ϭ͗N͘ M L ϭ0 , and, in conse- quence, the cancellation of Eq. ͑32͒.Therefore, we can conclude that the contribution from high frequencies to the term E AZA is zero or, at least, negligible.

͑33͒
At low frequencies (e ik ជ •r ជ j Ϸ1), р c ϵmc 2 ␣/ប, the integral in , from 0 to c , is quadratic in c and therefore in ␣.As a consequence, this contribution is of order ␣ 2 smaller than E AZA I ͑that is, of order ␣ 5 in g L (3) ͒ and may be neglected.
In order to calculate the high-frequency part, Ͼ c , we shall use the identity and a similar one for 3 /( n ϩ) 2 .As stated below Eq.
͑29͒, Ͼmc 2 ␣/ប and n , n Јϳmc 2 ␣ 2 /ប, hence each term in Eq. ͑34͒ makes a contribution of order ␣ with respect to the previous one.The main contribution to Eq. ͑33͒ can then be calculated by retaining only the first term of the righthand side of expression ͑34͒.In this case we can perform the sum over n and nЈ, and obtain

͑35͒
The contribution from the interaction of one electron with itself will be not considered here for the reasons explained above Eq.͑31͒.The terms in Eq. ͑35͒ where different electrons contribute lead to convergent integrals.Rearranging the terms in a way analogous to that in Eq. ͑31͒, for the operator in the first matrix element of Eq. ͑35͒ we obtain where only the even part in k ជ contributes to the angular integral.The integral in frequencies ͑or k͒ is straightforward extending the lower limit to zero ͓which gives a small error of order ␣ 2 E AZA I , as explained below Eq. ͑33͔͒.We obtain where which has rotational invariance.We shall show that the term E BZB is zero, but, before doing that, it is important to realize that it is of the same order as E AZA I ͑i.e., a contribution of order ␣ 3 to g L (3) ͒.In fact, r 12 is of the order of the Bohr radius, i.e., ប/(mc␣), so that R, given by Eq. ͑38͒, is proportional to ␣ 2 .We recall that G is proportional to ␣.This implies that the error of replacing the left-hand side of ͑29͒ by the first term is negligible ͑it would make a contribution of order ␣ 4 to g L (3) ͒.We see in Eq. ͑37͒ that E BZB is zero when is an eigenstate of L z ϩ2S z ϭJ z ϩS z , which occurs for the state ͉ a ͘ ϵ͉ 3 P 2 ,M J ϭ2͘.For the state ͉ b ͘ϵ͉ 3 P 1 ,M J ϭ1͘, an eigenstate of J z but not of S z , the contribution is also zero, as can be shown by inserting the identity ͚ J Ј M JЈ ͉LSJЈM J Ј ͗͘M J Ј JЈSL͉ between R and S z in Eq. ͑37͒, applying the Wigner-Eckart theorem, and taking into account that S z can neither change the configuration nor M J and that R cannot change J because of its rotational invariance.Then E BZB ϭ0, or is at least negligible.
We now analyze the term E AZB together with E BZA because the sum E AZB ϩE BZA allows an easier rearrangement of the operators.The term E AZB is given by and E BZA is given by a similar expression but with the appropriate change of the operators.The same considerations concerning the term E BZB apply here, and we discard n and n Ј compared with , and then sum in n and nЈ.We rearrange the operators in E AZB ϩE BZA following the same procedure as in Eqs.͑31͒ and ͑36͒, obtaining where

͑41͒
The operator T has rotational invariance, and we proceed in the same way as in E BZB to show that E AZB ϩE BZA ϭ0 or at least negligible.

V. CORRECTIONS FROM THE TERMS E AAZ ,E BAZ ,E ABZ ,E BBZ
The terms analyzed here are given by expression ͑7͒, where H l is H Z ͑which does not connect different radiation states͒ and H i and H j are H A or ͑and͒ H B , which verify ͗;0͉H A ͉;0͘ϭ0, and ͗;0͉H B ͉;0͘ϭ0.
It is now more convenient to consider the operator A ជ vac expanded in spherical waves instead of plane waves as in Eq. ͑5͒.The radiation states are then characterized by ͉l,m l ,k ជ ,⑀ ជ ͘, where l and m l are the photon angular momen- tum and its projection, respectively.The expression of E AAZ , for example, is given by where the state ͉n͘ can be any atomic eigenstate, including , while ͉nЈ͘ must be different from .We see in Eq. ͑42͒ that E AAZ is zero if is an eigenstate of J z ϩS z , since we obtain ͗nЈ͉͘ϭ0.This is the case for ͉ a ͘ϵ͉ 3 P 2 ,M J ϭ2͘.
For ͉ b ͘ϵ͉ 3 P 1 ,M J ϭ1͘, since S z can neither connect dif- ferent configurations nor different M J , there is only one state ͉nЈ͘, with the same configuration as b and having J ϭ2 and M J ϭ1.The sum over nЈ then consists of only one term.We denote by ͉J,M J ͘ the angular part of the state ͉n͘, and note that ͉J,M J ,l,m l ͘ϭ͚ J Ј ,M JЈ CЈ͉J,l,JЈ,M J Ј ͘ ͑CЈ is a Clebsch-Gordan coefficient͒.Taking into account the rotational invariance of p ជ •A ជ vac , which cannot connect different J, and applying the Wigner-Eckart theorem, we find that a product of two Clebsch-Gordan coefficients appears to come from the first and second matrix elements of Eq. ͑42͒.For each k ជ and each atomic configuration of ͉n͘, we must calcu- late the sum ͚ J,M J ,l,m l ͗Jϭ1,M J ϭ1͉J,M J ,l,m l ͘ ϫ͗J,M J ,lϪm l ͉Jϭ2,M J ϭ1͘, ͑43͒ which is zero due to the orthogonality of the Clebsch-Gordan coefficients, and then E AAZ ϭ0.
The operator S ជ •B ជ vac also has rotational invariance, and we can follow exactly the same procedure as before to show that E BAZ ϭ0, E ABZ ϭ0, and E BBZ ϭ0.

VI. CONCLUSIONS
We have analyzed those radiative corrections in the Zeeman effect of helium, which have not been considered in previous work.The atomic states are 2 3 P 1 and 2 3 P 2 .
These corrections are the third-order perturbative contributions from the Zeeman Hamiltonian and vacuum radiative interaction (Ϫe/m)(p ជ •A ជ vac ϩS ជ •B ជ vac ), retaining the terms linear in the external magnetic field and proportional to e 3 .
We have found that the greatest correction comes from terms proportional to L ជ •B ជ 0 and p ជ •A ជ vac , contributing to the orbital gyromagnetic factor with a correction of the same order (ϳ␣ 3 ) as that calculated by Anthony and Sebastian ͓2͔ in the second order of perturbation.Specifically, we have found that the correction is just one half that calculated by these authors, i.e., 1 2 ϫ2.39ϫ10Ϫ7 ͓see Eq. ͑25͔͒.Therefore, the result does not resolve the discrepancy between theory and experiment.We have also shown that the third-order corrections coming from terms involving S ជ •B ជ vac , are zero or negligible ͑of order ␣ 4 or lower͒.
ReE AZB ϩ2 ReE AAZ ϩ2 ReE BAZ ϩ2 ReE ABZ ϩ2 ReE BBZ , ͑8͒ where we used E BZA ϭE AZB * , E ZAA ϭE AAZ * , E ZAB ϭE BAZ * , E ZBA ϭE ABZ * , and E ZBB ϭE BBZ * .The corresponding third-order corrections of the gyromagnetic factors g ••• .Retaining only the first term of this expansion, expression ͑26͒ results equal to Ϫ1.The integral over frequencies in the second term of E AZA II